Asymptotic identification uncertainty of well-separated modes in operational modal analysis with multiple setups

Abstract

Operational modal analysis (OMA) aims at identifying structural modal properties with (output-only) ambient vibration data. In the absence of loading information, the identification (ID) uncertainty of modal properties becomes a valid concern in quality control and test planning. One recent development that addresses this aspect is ‘uncertainty law’, which aims at understanding how ID uncertainty depends on test configuration. Mathematically, uncertainty laws in OMA are asymptotic expressions for the ‘posterior’ (i.e., given data) variance of modal parameters. Analogous to the laws of large numbers in statistics, they are often derived assuming long data, small damping, and high signal-to-noise ratio. Following a Bayesian approach, this work develops the uncertainty law for OMA with multiple setup data, a common strategy to produce a ‘global’ mode shape covering a large number of locations with a small number of sensors in individual setups. It advances over previous results for single setup data, and is motivated by questions, e.g., how does the quality of global mode shape depend on sensor locations and setup schedule? Focusing on the case of fixed reference and distinct rovers, analytical study of the eigenvalue properties of mode shape covariance matrix reveals characteristic spatial patterns where principal uncertainty takes place, which can be of local or global nature. The theory is validated with synthetic, laboratory and field test data. By virtue of the Cramer-Rao bound, up to the same modeling assumptions, the uncertainty law dictates the achievable precision limit of OMA regardless of identification method.